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I'm looking for a good reference that has a detailed treatment of obstruction theory in the case where the target space is not simple. The specific situation I am interested in involves lifting a map of 3-skeletons from a $K(G, 1)$ to an arbitrary homotopy 3-type $X$ to the 4-skeletons (and hence to a true map between the spaces); the obstruction to this "should" live in $H^4(G, \pi_3(X))$ (with the appropriate action of $G$ on $\pi_3(X)$) via a local coefficient system, but I've been having trouble hashing out the details. Experts I've asked have given answers ranging from saying that it's impossible to saying that they're certain that it's possible but they don't know a reference that does it. Books I've looked at tend to gloss over the details, but they seem to indicate that this should work. Can anybody set me straight on this?

flag	asked Jul 9 2010 at 2:46 Evan Jenkins 4,146●1●8●21

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Trying to clarify what kind of obstruction you want. If $A$ and $B$ are $K(G,1)$ spaces with CW-decompositions, you have a map $f : A^3 \to B^3$ i.e. a map between the 3-skeleta. And you want to know if it extends to a map $f : A^4 \to B^4$ ? – Ryan Budney Jul 9 2010 at 2:56

My understanding is that A is a K(G,1) space and B is a space whose homotopy groups vanish above dimension 3. Suppose we have a map $f: A^3 \to B$. The problem then is to extend $f$ to a map from $A^4$ to $B$, and therefore from $A$ to $B$. – Gregory Arone Jul 9 2010 at 3:44

Sorry if this was unclear. The source space $A$ is $K(G, 1)$ (although this is probably inconsequential), and the target space $B$ (what I called $X$) is a connected homotopy 3-type (again, this is probably inconsequential). I want to extend a map between the 3-skeleta to the 4-skeleta, as you say, while possibly modifying the choice of 3-skeleton map (which itself was extended from the 2-skeleton in a non-unique way). It is this situation that, at least in the simple case, gives an obstruction in $H^4(A, \pi_3(B))$. – Evan Jenkins Jul 9 2010 at 3:48

Ah, okay, I hadn't heard the phrase "homotopy n-type" before. – Ryan Budney Jul 9 2010 at 5:46

Gregory Arone · Accepted Answer · 2010-07-09 03:19:05Z UTC

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Paul Olum developed some obstruction theory for maps into non-simple spaces back in the 1940-ies and 50-ies. You may want to check out his paper "Obstructions to extensions and homotopies", Annals of Mathematics, Vol 52, 1950, pp 1-50, if you have not looked at it yet.

edited Jul 9 2010 at 3:19

answered Jul 9 2010 at 3:12

Gregory Arone
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Thank you! The approach taken in this paper seems much nicer than the less general and more mysterious treatment (due originally to Whitehead?) that one sees in more modern references. – Evan Jenkins Jul 13 2010 at 7:25

Obstruction theory for non-simple spaces

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